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Section 6.1 - Sum and Difference Formulas Note: sin( A + B ) ≠ sin( A) + sin( B ) cos( A + B ) ≠ cos( A) + cos( B ) Sum and Difference Formulas for Sine, Cosine and Tangent sin( A + B ) = sin A cos B + sin B cos A sin( A − B ) = sin A cos B − sin B cos A cos( A + B ) = cos A cos B − sin A sin B cos( A − B ) = cos A cos B + sin A sin B tan( A + B ) = tan A + tan B 1 − tan A tan B tan( A − B ) = tan A − tan B 1 + tan A tan B 1 Example 1: Simplify each: a. cos( x + 60°) = π π b. sin x + − sin x − 4 4 2 3π Example 2: Given that tan( x) = 5 , evaluate tan x + 4 . 3 Example 3: Simplify each. a. sin10° cos 55° − sin 55° cos10° π 7π b. cos cos 12 12 c. tan 40° + tan 5° 1 − tan 40° tan 5° d. tan 80° − tan15° 1 + tan 80° tan15° π 7π + sin sin 12 12 4 Example 4: Find the exact value of each. (Hint: use sum/difference formulas) a. sin 15 0 7π b. cos 12 5 5π c. tan 12 6 Example 5: Suppose that sin α = 3 5 π and cos β = where 0 < α < β < . Find each of 5 13 2 these: a. sin(α + β ) b. cos(α − β ) 7 Example 6: Suppose cos α = 1 7 and tan β = − where π < α , β < 2π . Find 5 6 a. cos(α + β ) b. tan(α + β ) 8 Section 6.2 – Double and Half Angle Formulas Now suppose we are interested in finding sin(2 A) . We can use the sum formula for sine to develop this identity: sin(2 A) = sin( A + A) = sin A cos A + sin A cos A = 2 sin A cos A Similarly, we can develop a formula for cos(2 A) : cos(2 A) = cos( A + A) = cos A cos A − sin A sin A = cos 2 A − sin 2 A We can restate this formula in terms of sine only or in terms of cosine only by using the Pythagorean theorem and making a substitution. So we have: cos(2 A) = cos 2 − sin 2 A = 1 − 2 sin 2 A = 2 cos 2 A − 1 We can also develop a formula for tan(2 A) : tan(2 A) = tan( A + A) tan A + tan A = 1 − tan A tan A 2 tan A = 1 − tan 2 A These three formulas are called the double angle formulas for sine, cosine and tangent. 1 Double – Angle Formulas sin(2 A) = 2 sin A cos A cos(2 A) = cos 2 A − sin 2 A tan(2 A) = (Also: cos(2 A) = 2 cos 2 A − 1 = 1 − 2 sin 2 A ) 2 tan A 1 − tan 2 A Now we’ll look at the types of problems we can solve using these identities. 2 4 π Example 1: Suppose that cos α = − and < α < π . Find 7 2 a. cos(2α ) b. sin(2α ) c. tan(2α ) 3 Example 2: Simplify each: a. 2 sin 45° cos 45° b. cos 2 π 9 − sin 2 π 9 4 c. 2 tan15° 1 − tan 2 15° d. 1 − 2 sin 2 (6 A) 5 Half – Angle Formulas 1 − cos A A sin = ± 2 2 1 + cos A A cos = ± 2 2 sin A 1 − cos A A tan = = sin A 2 1 + cos A Note: In the half-angle formulas the ± symbol is intended to mean either positive or negative but not both, and the sign before the radical is determined by the quadrant in A which the angle terminates. 2 Now we’ll look at the kinds of problems we can solve using half-angle formulas. 6 Example 3: Use a half-angle formula to find the exact value of each. a. sin 15° 5π b. cos 8 7 7π c. tan 12 8 Example 4: Answer these questions for cos θ = 4 3π , < θ < 2π . 9 2 a. In which quadrant does the terminal side of the angle lie? b. Complete the following: ___ < θ 2 < ___ c. In which quadrant does the terminal side of θ 2 lie? θ d. Determine the sign of sin . 2 θ e. Determine the sign of cos . 2 θ f. Find the exact value of sin . 2 θ g. Find the exact value of cos . 2 θ h. Find the exact value of tan . 2 9 Section 6.3 - Solving Trigonometric Equations Next, we’ll use all of the tools we’ve covered in our study of trigonometry to solve some equations. An equation that involves a trigonometric function is called a trigonometric equation. Since trigonometric functions are periodic, there may be infinitely solutions to some trigonometric equations. Let’s say we want to solve the equation: The first angles that come to mind are: x = sin( x) = π 6 1 2 and x = 5π . 6 Remember that the period of the sine function is 2π ; sine function repeats itself after each rotation. Therefore, the solutions of the equation are: x = π 6 + 2kπ , x = 5π + 2kπ , where k is any 6 integer. Recall: For sine and cosine functions, the period is 2π . For tangent and cotangent functions, the period is π . 1 Example 1: a) Solve the equation in the interval [0,2π ) : 2 cos x = −1 b) Find all solutions to the equation: 2 cos x = −1 Example 2: a) Solve the equation in the interval [0, π ) : tan x = −1 b) Find all solutions to the equation: tan x = −1 2 Example 3: Solve the equation in the interval [0, π ) : 2sin(2 x) = 1 Example 4: Solve the equation in the interval [0,2π ) : csc 2 x = 4 3 Example 5: Find all solutions to the equation: cos(2 x) = 0 4 Example 6: Solve the equation in the interval [0,2π ) : 2 sin 2 x − 5 sin x − 3 = 0 5 Example 7: Solve the equation in the interval [0,2π ) : cos 2 x − 3sin x − 3 = 0 6 Example 8: Solve the equation in the interval [0,2π ) : cos(2 x) = 5 sin 2 x − cos 2 x 7 Example 9: Find all solutions to the equation: sin 2 x cos x = cos x 8 Example 10: Find all solutions: sec 2 x + 2 tan x = 0 9 Example 11: Solve the equation in the interval [0, 2) : cot(π x) = −1 Example 12: Find all solutions of the equation in the interval [0, 4π ) : x 2 sin = 1 2 10 Example 13: Find all solutions of the equation in the interval [0,2π ) : sec( x + 2π ) = 2 11 Example 14: Find all solutions of the equation in the interval [0, π ) : 3π 2 sin 2 x − = 2 2 12